In most periodic homogenization schemes one has to solve a so-called cell problem on a representative volume element, which can be put in the form of the Lippmann-Schwinger equation. The present study focuses on the singular value decomposition of the operator involved in the Lippmann-Schwinger equation, and the properties that can be derived from it, in the case of a conductivity problem. The decomposition is proposed for different choices of field representation bases: in real space, in Fourier space, or according to a sine and cosine decomposition, and it will be shown that these different decompositions can be deduced from each other using simple relationships. The singular values of the decomposition allow to simply find the convergence conditions of the fixed-point resolution scheme. The study of the fields corresponding to the singular vectors of the operator operator provides a better understanding of how the solution to a particular problem articulates the fields in the different phases of the material. The study of the projection on the basis of these eigenvectors of the solution of the problem and of the approximate solutions at different steps in the iterative resolution of the problem, shows that the singular values playing an important role in the solution are rather few in number. Indeed, it appears that after the first iteration of a fixed-point scheme, only the singular values corresponding to solutions “living” in the interface of the constituents still play a role, with all the others reaching their solution in the first iteration. These considerations make it possible to envisage solving strategies that focus on these singular values and, given their small multiplicity, would enable a much more efficient resolution, in terms of speed and precision.